\(\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx\) [237]

Optimal result

Integrand size = 29, antiderivative size = 257 \[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {d-c^2 d x^2}} \] Output:

-2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2) 
)/(-c^2*d*x^2+d)^(1/2)+2*I*b*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*polylog( 
2,-I*c*x-(-c^2*x^2+1)^(1/2))/(-c^2*d*x^2+d)^(1/2)-2*I*b*(-c^2*x^2+1)^(1/2) 
*(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))/(-c^2*d*x^2+d)^(1/2 
)-2*b^2*(-c^2*x^2+1)^(1/2)*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))/(-c^2*d*x^ 
2+d)^(1/2)+2*b^2*(-c^2*x^2+1)^(1/2)*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))/(- 
c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {a^2 \log (c x)}{\sqrt {d}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d}}+\frac {2 a b \sqrt {1-c^2 x^2} \left (\arcsin (c x) \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}+\frac {b^2 \sqrt {1-c^2 x^2} \left (\arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )-\arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )+2 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-2 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+2 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )}{\sqrt {d-c^2 d x^2}} \] Input:

Integrate[(a + b*ArcSin[c*x])^2/(x*Sqrt[d - c^2*d*x^2]),x]
 

Output:

(a^2*Log[c*x])/Sqrt[d] - (a^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d 
] + (2*a*b*Sqrt[1 - c^2*x^2]*(ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Lo 
g[1 + E^(I*ArcSin[c*x])]) + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2 
, E^(I*ArcSin[c*x])]))/Sqrt[d - c^2*d*x^2] + (b^2*Sqrt[1 - c^2*x^2]*(ArcSi 
n[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c* 
x])] + (2*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c*x 
]*PolyLog[2, E^(I*ArcSin[c*x])] - 2*PolyLog[3, -E^(I*ArcSin[c*x])] + 2*Pol 
yLog[3, E^(I*ArcSin[c*x])]))/Sqrt[d - c^2*d*x^2]
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.56, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5218, 3042, 4671, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx\)

\(\Big \downarrow \) 5218

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{c x}d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))^2 \csc (\arcsin (c x))d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 4671

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-2 b \int (a+b \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )\right )}{\sqrt {d-c^2 d x^2}}\)

Input:

Int[(a + b*ArcSin[c*x])^2/(x*Sqrt[d - c^2*d*x^2]),x]
 

Output:

(Sqrt[1 - c^2*x^2]*(-2*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])] + 
2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])] - b*PolyLog[3, - 
E^(I*ArcSin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c* 
x])] - b*PolyLog[3, E^(I*ArcSin[c*x])])))/Sqrt[d - c^2*d*x^2]
 

Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5218

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* 
(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* 
x^2]]   Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a 
, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.51

method	result	size
default	\(-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{\sqrt {d}}+\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}\)	\(387\)
parts	\(-\frac {a^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{\sqrt {d}}+\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}-\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}\)	\(387\)

Input:

int((a+b*arcsin(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-a^2/d^(1/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+b^2*(-c^2*x^2+1)^( 
1/2)*(-d*(c^2*x^2-1))^(1/2)*(arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))- 
arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*arcsin(c*x)*polylog(2,-I* 
c*x-(-c^2*x^2+1)^(1/2))+2*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2) 
)-2*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))+2*polylog(3,-I*c*x-(-c^2*x^2+1)^(1 
/2)))/d/(c^2*x^2-1)-2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(I*a 
rcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x 
^2+1)^(1/2))+polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-polylog(2,I*c*x+(-c^2*x^ 
2+1)^(1/2)))/d/(c^2*x^2-1)
 

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas" 
)
                                                                                    
                                                                                    
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^ 
2)/(c^2*d*x^3 - d*x), x)
 

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:

integrate((a+b*asin(c*x))**2/x/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral((a + b*asin(c*x))**2/(x*sqrt(-d*(c*x - 1)*(c*x + 1))), x)
 

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima" 
)
 

Output:

-a^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/sqrt(d) - sqr 
t(d)*integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*a 
rctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c 
^2*d*x^3 - d*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsin(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x\,\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((a + b*asin(c*x))^2/(x*(d - c^2*d*x^2)^(1/2)),x)
 

Output:

int((a + b*asin(c*x))^2/(x*(d - c^2*d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {2 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) a b +\left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) b^{2}+\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2}}{\sqrt {d}} \] Input:

int((a+b*asin(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

(2*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*x),x)*a*b + int(asin(c*x)**2/(sqr 
t( - c**2*x**2 + 1)*x),x)*b**2 + log(tan(asin(c*x)/2))*a**2)/sqrt(d)